### The second exam

Problem#1 Extra
credit
#2 #3 #4 #5 #6 #7 Total
Max grade 15 3 14 14 12 14 15 15 94
Min grade 5 0 0 1 1 2 2 0 35
Mean grade 12.28 1.81 8.81 7.5 8.5 7.31 8.25 11.69 66.16
Median grade 14 3 9.5 8 9 6 7.5 13 67.25

Numerical grades will be retained for use in computing the final letter grade in the course. Here are approximate letter grade assignments for this exam:

 Letterequivalent Range A B+ B C+ C D F [85,100] [80,84] [70,79] [65,69] [55,64] [50,54] [0,49]

An answer sheet for the exam is available. 16 students took the exam. The contrasts between Mean and Median scores describe overall good performance.

Problem 1 (15 points)
2 points for a sketch of the contour.
6 points for the residue computation.
6 points for showing that the ML estimate makes the integral over the "other" part of the contour go to 0.
1 point for getting the answer.
3 points for the correct extra credit answer.

Problem 2 (14 points)
a) 7 points. simply and simple each get 1 point, properly used. The equation itself gets 3 points. The other "stuff" gets 2 points.
b) 7 points. The answer, and some method for getting it.

Problem 3 (15 points)
The singularity at 0 can earn 7 points. The other singularities can earn 8 points.
At 0, 1 point for the statement that the singularity is removable and 1 point for the value of the residue. The other 5 points are for some supporting evidence.
Similar scoring will be used for the non-zero singularities: 1 point for the residue and 1 point for identifying the singularity as a pole and 1 point for giving the order of the pole. Again, supporting evidence can earn 5 points.

Problem 4 (12 points)
a) 8 points. Apply Liouville's Theorem (2 points) to a "correct" function (4 points) and get the conclusion (2 points).
b) 4 points. 2 for the assertion that the exponential function does not have modulus bounded away from 0. 2 more points for explaining why.

Problem 5 (14 points)
2 points for information about a power series for sin(z).
3 points for information about a power series for 1/(z-1)2.
5 points for combining them usefully (multiplication, division, etc.)
Then 4 points for each of the correct terms in the answer.
An alternative unrecommended direct approach is possible. So computing correctly f(k)(0) will earn k points, where k is an integer running from 1 to 4. Thus computing f(0) itself earns nothing (sigh). Assembling the terms in the Taylor series (with the factorials) earns, as above, 4 points for each of the correct terms in the answer.

Problem 6 (14 points)
a) 4 points. Many correct answers are possible, and if another answer (inferentially) needs to be chosen to answer d), that's o.k. 2 points for excluding 1, and 2 points for making a correct "cut" in C. b) 2 points for the correct answer.
c) 2 points for the correct answer.
d) 7 points. Some derivatives of sqrt(z) will earn 2 points. This should be a series in integer powers of z-1. Each correct term will earn 1 point.

Problem 7 (15 points)
5 points for instantiating the conventional "dictionary" changing this to a complex line integral. Finding the singularities of the integrand and manipulating it algebraically correctly earns another 5 points. Applying the Residue Theorem, and computing the correct residue and getting the correct answer earns the final 5 points. The not-uncommon error of failing to compensate for the fact that the dictionary response does not get a monic polynomial will lose 2 points (this usually yields an extra b in the answer).